Models for predicting extinction times: shall we dance (or walk or jump)?

Abstract

Measures of the risk of extinction, such as the expected time to extinction, are often used in population viability analysis (PVA) and in subsequent decision-making procedures, to gauge the impact of various management actions. Since PVA results are often used in the allocation of conservation funding, which in turn has a major impact on the future persistence of populations, the calculation of quantities such as the expected time to extinction is of great importance. These quantities are typically calculated by assuming a specific model for changes in the population over time. In particular, diffusion models are often used because they are simple to analyse and often give rise to explicit formulæ for most quantities of interest. However, while they are widely used, they frequently lead to inaccurate predictions of critical quantities such as the expected time to extinction. Hence, management decisions based on these predictions may be similarly flawed. Often, a more appropriate model for describing the behaviour of the population in question is a discrete-state Markov process describing the actual number of individuals in the population. The most commonly used such models are birth-death processes or extensions thereof which allow for catastrophic events. Unfortunately, whilst these may be more appropriate for modelling the dynamics of the population in question, they are usually more difficult to work with, from both analytical and computational points of view. For these reasons, it is important to find some balance between accuracy of predictions based on the models and tractability of the method of prediction. Advances can be made by considering the limiting processes that correspond to these discrete-state models; in particular, Ornstein-Uhlenbeck processes and piecewise-deterministic processes with stochastic jumps. These models may still provide inaccurate predictions of extinction times, but should show improvement over the simplest Brownian motion approximation. We consider populations that have density dependent demographic rates (in a specially-defined sense), and which may also be subject to environmental catastrophes. In particular, we assume that these populations may be modelled by continuous-time Markov chains - the stochastic SIS logistic model with or without binomial catastrophes occurring at a constant rate - and compare the accuracy of several approximations to the expected time to extinction. We contrast the various advantages of several methods for predicting extinction times for the above mentioned models, and compare these predictions for simulated data and a population of Bay checkerspot butterflies using model parameters estimated from data. We pay particular attention to the question of whether the extra analytical and computational effort required for the more complex models is necessary to inform decision-making in a conservation context. We find that a variety of different models may give comparable results for measures of the risk of extinction (such as the expected time to extinction). This is true even in the situations we examine where catastrophes are known to play a role in population dynamics, but are not modelled when analysing the data. One model we consider in detail, the stochastic SIS logistic model and its Ornstein-Uhlenbeck diffusion approximation, is particularly robust in allowing for either strong or weak limiting of populations by their carrying capacities, and in adjusting for catastrophic events. A simpler geometric Brownian motion approximation may also provide reasonable results, but is less reliable due to shortcomings in its estimation of the population ceiling. Finally, we determine empirically that heuristic approximation methods for the stochastic SIS logistic model subject to catastrophes can provide accurate values for the expected time to extinction when the true parameters are known. The present lack of a suitable estimation procedure for these models would preclude their wider use, but fortunately other models, such as the Ornstein-Uhlenbeck approximation, can provide reasonable estimates of the expected time to extinction.